Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

120 2 Null-Field Method

Table 2.2. Occupation of the global matrix

For distributed sources, the identity matrix I is replaced by the Q13 matrix

orders corresponding to the subsequent layers are in descending order. In contrast to these prescriptions, the solution method using a recurrence relation for T -matrix calculation requires all matrices to be of the same order, i.e.,

This requirement implies that the same number of basis functions must be used to approximate the surface fields on each layer. For concentrically layered spheres, this requirement is not problematic because the basis functions are orthogonal on spherical surfaces. For nonspherical layered particles, we approximate the surface fields by a complete system of vector functions and it is natural to use fewer basis functions for smaller layer surfaces. However, for convergence tests it is simpler to consider a single truncation index [181, 248].

2.5.3 Formulation with Discrete Sources

For a two-layered particle as shown in Fig. 2.2, the null-field equations formulated in terms of distributed vector spherical wave functions (compare to (2.70), (2.71) and (2.72))

122 2 Null-Field Method

0−Ql−1(ki,l−1, l, ki,l−1, l − 1)

pq

The expressions of the elements of the Ql matrix are given by (2.84)–(2.87) with the localized vector spherical wave functions replaced by the distributed vector spherical wave functions. The transition matrix is

where the Q1 (ks, 1, ki,1, 1) and Q1 (ks, 1, ki,1, 1) matrices contain as rows

matrix contains as rows and columns the vectors M3ν (ksr1), Nν3(ksr1) and M 1µ(ksr1), N 1µ(ksr1), respectively.

The use of distributed vector spherical wave functions improves the numerical stability of the null-field method for highly elongated and flattened layered particles. Although the above formalism is valid for nonaxisymmetric particles, the method is most e ective for axisymmetric particles, in which case the z-axis of the particle coordinate system is the axis of rotation. Applications of the null-field method with distributed sources to axisymmetric layered spheroids with large aspect ratios have been given by Doicu and Wriedt [50].

2.5.4 Concentrically Layered Spheres

For a concentrically layered sphere, the precedent relations simplify to those obtained in the framework of the Lorenz–Mie theory. In this specific case, we use the recurrence relation (2.107) with Ql given by (2.106). All matrices are diagonal and denoting by (Tl,l+1,...,N )1n and (Tl,l+1,...,N )2n the elements of the matrix T , we rewrite the recurrence relation as

(2.117)

where

is the relative refractive index of the layer l with respect to the layer l − 1. To obtain a stable scheme for computing the T matrix, we express the above recurrence relation in terms of the logarithmic derivatives An and Bn:

where pn and sn are given by (2.118) and (2.119), respectively, and rn and qn are now given by

1 Bn (mr,lxl) h(1)n (mr,lxl)

rn = 1 + (Tl+1,l+2,...,N )n An (mr,lxl) jn (mr,lxl)

and

2 Bn (mr,lxl) h(1)n (mr,lxl)

qn = 1 + (Tl+1,l+2,...,N )n An (mr,lxl) jn (mr,lxl) ,

respectively.

The computation of Lorenz–Mie coe cients for concentrically layered spheres has been considered by Kerker [115], Toon and Ackerman [222], and Fuller [76], while recursive algorithms for multilayered spheres have been developed by Bhandari [13] and Mackowski et al. [154].

124 2 Null-Field Method

2.6 Multiple Particles

In this section we extend the null-field method to the case of an arbitrary number of particles by using the translation properties of the vector spherical wave functions. Taking into account the geometric restriction that the particles do not overlap in space, we derive the expression of the transition matrix for restricted values of translations. Our treatment closely follows the original derivation given by Peterson and Str¨om [187, 188].

2.6.1 General Formulation

For reasons of clarity of the presentation we first consider the generic case of two homogeneous particles immersed in a homogeneous medium with a relative permittivity εs and a relative permeability µs. The scattering geometry is depicted in Fig. 2.5. The surfaces S1 and S2 are defined with respect to the particle coordinate systems O1x1y1z1 and O2x2y2z2, respectively, while the coordinate system of the ensemble or the global coordinate system is denoted by Oxyz. The coordinate system O1x1y1z1 is obtained by translating the coordinate system Oxyz through r01 and by rotating the translated coordinate system through the Euler angles α1, β1 and γ1. Similarly, the coordinate system O2x2y2z2 is obtained by translating the coordinate system Oxyz through r02 and by rotating the translated coordinate system through the Euler angles α2, β2 and γ2. The main assumption of our analysis is that the smallest circumscribing spheres of the particles centered at O1 and O2, respectively, do not overlap. The boundary-value problem for the two scattering particles depicted in Fig. 2.5 has the following formulation.

Given the external excitation Ee, He as an entire solution to the Maxwell equations, find the scattered field Es, Hs and the internal fields Ei,1, Hi,1

Fig. 2.5. Geometry of two scattering particles